Generalizing the Geometry of Model Merging Through Frechet Averages
Authors: Marvin F. da Silva, Mohammed Adnan, Felix Dangel, Sageev Oore
Organizations: Faculty of Computer Science, Dalhousie University · 2Vector Insitute for Artificial Intelligence · Schulich School of Engineering, University of Calgary · Department of Computer Science and Software Engineering, Concordia University · Mila - Quebec AI Institute
Abstract
Model merging aims to combine multiple models into one without additional training. Naïve parameter-space averaging can be fragile under architectural symmetries, as their geometry does not take them into account. In this work we show that not only the geometry, but also the averaging procedure itself, must be symmetry-invariant to achieve symmetry-aware merges. Consequently, we propose a general solution: merging as Fréchet averaging, i.e., selecting parameters that minimize a sum of geodesic distances on an appropriate manifold. In this view, the key design choice is the overall geometry, i.e., the choice of metric, manifold, and distance approximation, that determines what it means for two models to be "close". We show that Fréchet averaging, combined with simplifying assumptions, contains Fisher merging. Building on this, we examine the particular case of low-rank adapters (LoRA), whose symmetries induce a distinct geometry: that of a quotient manifold. We outline the limitations of current LoRA merging methods, propose a practical algorithm for this setting, and show how they compare with other commonly used approaches.
Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.
Model merging combines fine-tuned checkpoints into a single multi-task model without retraining. Existing methods - such as task arithmetic, model soups, TIES, and DARE - are computationally efficient and empirically successful, but rely on heuristic design choices and lack formal optimality guarantees. We show that merging can be formulated as a convex quadratic programme over residual updates, yielding weights that minimise a squared-output calibration objective using calibration inputs and fine-tuned model outputs, and subsuming existing methods as special cases. Our framework yields a closed-form diagnostic - the fraction of residual energy captured by a chosen basis - that predicts downstream merge quality using only the calibration set. Empirically, the QP matches or outperforms existing methods in the single-layer setting, and we characterise when the optimal basis provides significant gains over the cheaper diagonal QP. We extend to multi-layer merging via a sequential layer-wise algorithm and demonstrate consistent gains across language and vision benchmarks.
Bethan Evans, Benjamin Etheridge, Stephen Roberts +1
LoRA adapters provide an efficient way to specialize a pretrained model for many downstream tasks, but deploying one adapter per task requires adapter storage and task selection at inference time. Model merging addresses this issue by combining independently trained adapters into one multi-task adapter. Recent SVD-based LoRA merging methods mainly focus on constructing shared or task specific directions, while the coefficients assigned to the final directions are often directly from the original task SVD. On a fixed merged basis, inherited coefficients preserve component order with high rank correlation, yet their magnitudes differ substantially from the coefficients induced by the task updates. To address this mismatch, we propose CT-Merging, a LoRA-aware merging algorithm that estimates consensus directions from average task subspace projectors and assigns task-level RMS coefficient scales in the final update. CT-Merging uses repeated support across task SVD subspaces to construct the common basis, while reducing reliance on rank wise SVD magnitudes after direction construction. On the DC-Merge CLIP adapter benchmark, CT-Merging achieves superior average normalized accuracy compared to state-of-the-art merging methods and further improves over DC-Merge by 2.56 points on ViT-B/32 and 1.51 points on ViT-L/14 KnoTS-trained checkpoints.